Coactions of weak multiplier Hopf algebras
Abstract
In this article, we investigate the notion of a coaction for a weak multiplier Hopf algebra. We show that there is a duality for actions and coactions of weak algebraic quantum groups. Then we obtain the well known Blattner-Montgomery duality theorem for weak algebraic quantum groups in the framework of dual pairs. We form a Morita context when the weak multiplier Hopf algebra admits a two-sided faithful integral and show that the coaction is Galois if and only if the map in the Morita context is surjective. As an application, we obtain the result that a grading by any groupoid G can be considered as a CG-comodule algebra. We also construct a Morita context connecting the smash product A# C(G) and the coinvariant subalgebra AC(G). Finally we construct an isomorphism between (R# C(G))# CG and R⊗ (C(G)# CG) when R is graded by G.
Related
- Partial coactions of Hopf algebras on nonunital algebras and categories
- Hopf Images of Hopf algebra Coactions
- Exact sequences of representation categories of weak Hopf algebras
- Bohr sets for multiplier algebras of complete Pick spaces
- Hochschild cohomology and AS-Gorenstein property of weak Hopf Galois extensions
BibTeX
@misc{zhou2026coactions,
title = {{Coactions of weak multiplier Hopf algebras}},
author = {Nan Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24406},
eprint = {2610.24406},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24406},
}
RIS
TY - UNPB
TI - Coactions of weak multiplier Hopf algebras
AU - Nan Zhou
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24406
UR - https://arxiv.org/abs/2610.24406
AB - In this article, we investigate the notion of a coaction for a weak multiplier Hopf algebra. We show that there is a duality for actions and coactions of weak algebraic quantum groups. Then we obtain the well known Blattner-Montgomery duality theorem for weak algebraic quantum groups in the framework of dual pairs. We form a Morita context when the weak multiplier Hopf algebra admits a two-sided faithful integral and show that the coaction is Galois if and only if the map in the Morita context is surjective. As an application, we obtain the result that a grading by any groupoid G can be considered as a CG-comodule algebra. We also construct a Morita context connecting the smash product A# C(G) and the coinvariant subalgebra A^C(G). Finally we construct an isomorphism between (R# C(G))# CG and R⊗ (C(G)# CG) when R is graded by G.
ER -